#606762

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,14,17,18)(15,19,16,20)(21,22,27,25)(23,28,24,31)(26,33,34,29)(30,36,32,35);(6,8)(9,12)(14,18)(15,16)(21,23,27,24)(22,28,25,31)(26,30,34,32)(29,35,33,36);(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,24,27,23)(22,28,25,31)(26,32,34,30)(29,35,33,36);(5,8,10,6)(7,12,11,9)(13,15,17,16)(14,19,18,20)(21,25,27,22)(23,28,24,31)(26,33,34,29)(30,35,32,36);(5,9,10,12)(6,11,8,7)(13,16,17,15)(14,20,18,19)(21,26)(22,29)(23,30)(24,32)(25,33)(27,34)(28,35)(31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 513958);